Quadratic+Functions

= __Quadratic Functions__=

//*y is sometimes written as f(×) and f(x) is function form*// A quadratic function is an equation in which the largest exponent is 2.
 * //What is a quadratic function?//**

The graph of a quadratic function is a parabola whose major axis is parallel to the y- axis.
 * //What does a quadratic function look like?//**


 * f(x)=x²-x-2**

//**What does a quadratic equations look like?**//

A quadratic equation looks like the graph of a parabola which is:

The a in the equation determines the size and the direction of the parabola. If a is positive then the parabola opens upward and if it's negative it opens downward.
 * y=ax²+bx+c**-->Standard Form
 * f(x)=ax²+bx+c** and also,
 * y=a(x-h)²+k** where //(h,k)// is the vertex of a parabola.

For any function in that form, the x intercepts can be determined by the quadratic formula which is:


 * //How would you graph a quadratic function by hand?//**

To graph a quadratic equation by hand, first you have to identify the x and y intercepts and the vertex. To do this, you put the equation into standard form (y=ax²+bx+c). To find the y intercept, set x to 0 and solve for y. To find the x intercept, set the y to 0 and use the quadratic formula or factor it out. Once you find them, you can put the equation into graphing form and the you must find the vertex. See the Parabolas, Completing the Square, and Averaging the X-intercepts page for more info.

From there, once you have found the intercepts and the vertex you can graph it and know that it should be in the shape of a parabola.


 * //Where are quadratic functions represented in everyday life?//**

Something can be modeled by a quadratic function is a function that gives position as a function of time for motion with constant acceleration. Our favorite constant acceleration is gravity. So things falling due to gravity can be modeled by quadratic functions.


 * //What are some characteristics of quadratic functions?//**


 * Graph shaped like a "U"
 * All quadratic functions are similar

**f(x)=(x-1)(3x+4)**

To put an equation like the one above into standard form, multiply the first number in the first set of parenthesis by the first number of the second set of parenthesis, then multiply the second number of the first parenthesis by the second number of the second set of parenthesis. Then multiply the second number of the first set of parenthesis by the first number of the second set of parenthesis. (FOIL=First. Outter. Inner. Last)

The final product will be **3x²+x-4 **

Example of how to solve for x and y intercepts: Find the y intercept for the equation y=x²+4x-12 If x=0, then y=(0)²+4(0)-12=-12 The y intercept is (0,-12).

Use the quadratic formula to solve for x. If ax²+bx+c=0,

math \frac{-b\pm\sqrt{b^2-4ac}}{2a} math a = 1 b = 4 c = -12

Using the formula gives x = 2 or x = -6 So the x-intercepts are (2,0) and (-6,0).

-line of symmetry - Its the line that is between the parabola like the X or the Y axis -vertex- Vertex is the place where the parabola turns, which is also called the turning point. Line of Symmetry and Vertex share a common relationship in between. they both are somewhat related. All parabolas are symmetrical and they intersect at the point known as the vertex, where both ends of the parabolas meet.